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Essential supremum norm differentiability. (English) Zbl 0593.46016

The main results of this paper are the following
Theorem 3.1. Let \((\Omega,\Sigma,\mu)\) be a finite measure space, X a Banach space, and \(f\in L_{\infty}(\mu,X)\) with \(f\neq 0\). Then f is a smooth point of \(L_{\infty}(\mu,X)\) iff there exists an atom \(A_ 0\) for \(\mu\) such that
(1) \(\| f\| >ess \sup \{\| f(\omega)\|:\omega \in \Omega \setminus A_ 0\}\), and
(2) \(x_ 0\) is a smooth point of X, where \(x_ 0\) is the essential value of f on \(A_ 0.\)
Corollary 3.2. The norm on \(L_{\infty}(\mu,X)\) is Fréchet differentiable at f iff there exists an atom \(A_ 0\) for \(\mu\) such that (1) \(\| f\| >ess \sup \{\| f(\omega):\omega \in \Omega \setminus A_ 0\}\), and
(2) the norm on X is Fréchet differentiable at \(x_ 0.\)
An application of these results is given to the space of all bounded linear operators from \(L_ 1(\mu,{\mathbb{R}})\) into X.
Reviewer: M.Kadets

MSC:

46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
47B38 Linear operators on function spaces (general)
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